Holographic Phases of Renyi Entropies

TL;DR

This work analyzes Rényi entropies $S_n$ of d-dimensional CFTs with a spherical entangling surface via the AdS/CFT correspondence, mapping $S_n$ to the thermal entropy of a CFT on hyperbolic space ${\mathbb H}_{d-1}$. The main finding is that a phase transition in $S_n$ can occur as the Renyi parameter $n$ is varied when the theory contains a sufficiently light scalar operator, with the transition triggered by hair formation on hyperbolic black holes and the critical $n_c$ depending on the lowest operator dimension $\Delta$. The authors provide analytic near-horizon arguments and numerical results in 5D and linearized analyses for normalizable modes to show that hyperbolic black holes become unstable to scalar hair at low temperatures, yielding non-analyticities in $S_n$ that reflect the entanglement spectrum via the function $d(\lambda)$. These results connect holographic black hole physics to CFT data such as the BF bounds and $\Delta$, highlighting that Rényi entropies can exhibit non-analytic behavior away from $n=1$ in large-$N$ theories and offering a concrete bulk mechanism for the entanglement spectrum structure.

Abstract

We consider Renyi entropies of conformal field theories in flat space, with the entangling surface being a sphere. The AdS/CFT correspondence relates this Renyi entropy to that of a black hole with hyperbolic horizon; as the Renyi parameter $n$ increases the temperature of the black hole decreases. If the CFT possesses a sufficiently low dimension scalar operator the black hole will be unstable to the development of hair. Thus, as $n$ is varied, the Renyi entropies will exhibit a phase transition at a critical value of $n$. The location of the phase transition, along with the spectrum of the reduced density matrix $ρ$, depends on the dimension of the lowest dimension non-trivial scalar operator in the theory.

Figures (7)

• Figure 1: Thermal entropy $S_{therm}$ as a function of the temperature $T$. The upper (red) curve is for the unbroken phase (Einstein black hole). The lower (light red, orange, light green, green, blue-green and blue) curves are for hairy black holes with $\mu^2L^2= -2.2, -3,-3.5,-3.75, -3.9375, -4$ ($\Delta=3.34,3,2+1/\sqrt{2},2.5, 2.25, 2$). The critical temperatures are $T_c=0.0037,~0.020, ~0.045,~0.070, 0.106, 1/2\pi$. Note the lower right (blue) curve is for the scalar at the AdS$_5$ BF bound, for which the critical temperature is that of the massless black hole.
• Figure 2: Rényi entropy as a function of $n$ for $\mu^2L^2=-1,-4,-3.75$ and $-3.9375$. The phase transition is at $n_{crit}\simeq 2.28$ for $\mu^2L^2=-3.75$ and $1.52$ for $\mu^2L^2=-3.9375$.
• Figure 3: The spectral function $d(\lambda)$ for the cases $\mu^2 L^2 = -1$, where the black hole is always Einstein, and $\mu^2 L^2 = -4$, where the black hole always has scalar hair. There are delta functions at the lowest eigenvalues, $\lambda^E_1=0.855$ for the Einstein (red line) and $\lambda^{ES}_1=0.911$ for the Einstein-scalar (blue line). For the pure Einstein case and with $V_H=G_N=L=1$, we obtain the same result as Hung:2011nu.
• Figure 4: Lowest eigenvalue of $\rho$ as a function of conformal dimension.
• Figure 5: Critical temperature as a function of conformal dimension.